Switched-capacitor (SC) DC-DC converters are a category of DC-DC converters which are comprised of capacitors and switches and contain no magnetic elements, such as inductors and transformers. As a result, switched-capacitor converters are particularly well-suited for monolithic integration and are most commonly used in low-power applications.
Unlike traditional, magnetics-based DC-DC converters, the ideal, unloaded conversion ratio, Mi, of a switched-capacitor DC-DC converter is entirely determined by its structure. For a given topology, the output voltage is unable to exceed the ideal value Vout=Mi·Vin. The maximum attainable efficiency of all SC converters depends on this ideal conversion ratio and is given as
                              η                      ma            ⁢                                                  ⁢            x                          =                              1                          M              i                                ·                                                    V                out                                            V                                  i                  ⁢                                                                          ⁢                  n                                                      .                                              (        1        )            
It is apparent from (1) that the efficiency is highest when the converter is operating close to the ideal value, i.e., Vout=Mi·Vin. Thus, SC converters are most effective in applications where the line does not vary significantly. It is also evident that the only way to extend the input voltage range and fundamentally improve the efficiency of switched-capacitor DC-DC converters is by utilizing a variable, multi-gain SC converter configuration.
The attainable ideal conversion ratios of a two-phase switched-capacitor DC-DC converter are known and can be expressed as
                              M          i                =                              A            ⁡                          [              N              ]                                            B            ⁡                          [              N              ]                                                          (        2        )            where A[N] and B[N], are integer values that are bounded by the number of floating capacitors N. These boundaries are given as1≤P[N],Q[N]≤FN+1  (3)where FN is the N-th Fibonacci number. The Fibonacci numbers follow the sequenceFN+1=FN+FN−1  (4)where F1=1 and F0=1.
Thus, with an increased number of capacitors, an SC converter is capable of attaining an increased number of ideal conversion ratios. Nevertheless, there is a strong motivation to limit the number of capacitors in a SC converter. While a large number of active devices can be integrated in a small area using modern CMOS processes, current integrated capacitors typically achieve densities of 0.1-10 nF/mm2. This low capacitive density significantly increases the size of the converter and thus its cost. In addition, integrated capacitors exhibit high parasitic bottom-plate capacitances, degrading the efficiency and performance.
A variety of switched-capacitor DC-DC converters have been proposed. These include those found in U.S. Pat. No. 5,414,614 issued to Motorola on May 9, 1995, U.S. Pat. No. 6,198,645 issued to National Semiconductor Corporation on Mar. 6, 2001, U.S. Pat. No. 8,259,476 issued to Ben-Yaakov et al on Jul. 29, 2009, U.S. Pat. No. 8,817,501 issued to Arctic Sand Technologies Inc. on Aug. 26, 2014, and U.S. Pat. No. 9,362,818 issued to Rf Micro Devices Inc. on Jun. 7, 2016. These prior converters, however, can generate only a limited number of ideal conversion ratios. In some cases, these converters cannot both step-up and step-down input voltage and, in other cases, use an excessively higher number of capacitors and switching states than are required.